{
  "abstract": "We study the mixed local and nonlocal Kirchhoff problem $$\\displaylines{ M\\Big(\\int_{\\Omega}|\\nabla u|^p\\,dx+ \\iint_{\\mathbb R^{2N}}\\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\\,dx\\,dy\\Big) \\bigl(-\\Delta_pu+(-\\Delta)^s_pu\\bigr) =f(x,u)+\\lambda|u|^{p^*-2}u, \\quad \\text{in }\\Omega,\\cr u=0,\\quad \\text{in }\\mathbb R^N\\setminus\\Omega, }$$ where $\\Omega\\subset\\mathbb R^N$ is a bounded smooth domain, $N\\geq 3$, $0< s< 1< p< N$, $p^*=Np/(N-p)$ and $\\lambda\\geq 0$. $M$ is a continuous function that vanishes at the origin. $f$ is a Caratheodory function that is odd in its second variable. When $M(t)$ is comparable to $t^{\\alpha-1}$ with $1<\\ alpha< p^*/p$, Krasnosel'skii genus theory gives infinitely many pairs of nontrivial weak solutions for $\\lambda=0$ under subcritical growth assumptions on $f$. If the lower-order growth exponent exceeds $p$, we combine a truncation with Moser iteration. For each positive integer $k$, we obtain at least $k$ distinct pairs of nontrivial weak solutions when $\\lambda$ is sufficiently small, with the upper bound on $\\lambda$ depending on $k$. In the Kirchhoff critical case $\\alpha=p^*/p$, we prove the Palais-Smale condition and obtain infinitely many pairs of nontrivial weak solutions for $0\\leq \\lambda< AS^{p^*/p}$, where $A$ is the lower bound coefficient of $M$ and $S$ is the mixed Sobolev constant.",
  "abstract_shown": true,
  "affiliations": [
    "Southwest Univ., Chongqing, China"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 2,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Wenjing Chen"
    },
    {
      "affs": [
        0
      ],
      "name": "Yunchuan Dai"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0fe6fr5zxm4p4.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": "http://ejde.math.txstate.edu/Volumes/2026/80/abstr.html",
  "id": "0fe6fr5zxm4p4",
  "issns": [
    "1072-6691"
  ],
  "landing_url": "http://ejde.math.txstate.edu/Volumes/2026/80/abstr.html",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "lcc",
      "QA1-939"
    ]
  ],
  "openalex": null,
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": "Texas State University",
  "ris": "https://research.draconitic.com/paper/0fe6fr5zxm4p4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T13:44:10Z",
      "native_id": "f639727b2c6b45f19eec73773f70b3f4",
      "source": "doaj",
      "url": "http://ejde.math.txstate.edu/Volumes/2026/80/abstr.html"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "lcc:QA1-939",
      "subfield": "general"
    }
  ],
  "title": "Existence and multiplicity of solutions for mixed local and nonlocal Kirchhoff type problems",
  "title_text": "Existence and multiplicity of solutions for mixed local and nonlocal Kirchhoff type problems",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T13:44:10Z",
  "url": "https://research.draconitic.com/paper/0fe6fr5zxm4p4",
  "venue": "Electronic Journal of Differential Equations",
  "versions": []
}